428 research outputs found
Sharp weighted estimates for approximating dyadic operators
We give a new proof of the sharp weighted inequality ||T||_{L^2(w)}
\leq c [w]_{A_2} where is the Hilbert transform, a Riesz transform, the
Beurling-Ahlfors operator or any operator that can be approximated by Haar
shift operators. Our proof avoids the Bellman function technique and two weight
norm inequalities. We use instead a recent result due to A. Lerner to estimate
the oscillation of dyadic operators.Comment: To appear in the Electronic Research Announcements in Mathematical
Science
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Uniform Rectifiability and Harmonic Measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains
Let , , be a closed, Ahlfors-David regular
set of dimension satisfying the "Riesz Transform bound"
Assume further that is the boundary of a domain
satisfying the Harnack Chain condition plus an
interior (but not exterior) Corkscrew condition. Then is uniformly
rectifiable
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